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Stabilisation of spin-singlet electronic states in transition metal clusters with a delocalised electron pair
THEO CHEM ELSEVIER
Journal
of Molecular
Structure
(Theochem)
330 (1995)
139-143
Stabilisation of spin-singlet electronic states in transition metal clusters with a delocalised electron pair S.A. Borshch Received
24 September
1993; accepted
27 September
1993
Abstract
A modelis proposedfor the interpretation of magneticpropertiesof polynuclearmixed-valencecomplexescontaining a pair of delocalisedelectrons.It is shownthat in many casesthe delocalisationleadsto a spin-singletground stateof clusters.The combineddelocalisation+ vibronic mechanismof stabilisationof the spin-singletstate is proposedfor clusterscontaining trianglesof metal atoms.
1. Introduction The magnetic properties of molecular systems containing paramagnetic ions in most cases are determined by localised unpaired electrons. Polynuclear clusters of transition metals present analogies, on a molecular level, to solid-state magnetic dielectrics. However, over the past few years the synthetic chemistry has provided more and more molecular systems in which the unpaired electrons can migrate between metallic centres. First of all we mean the mixed-valence compounds containing atoms of the same metal in different valence states. If the surroundings of metal atoms are equivalent chemically and structurally, the conditions are especially favourable for the delocalisation of the excess electrons. Thus mixed-valence clusters can be considered as molecular analogues of solids with itinerant electrons. Until now the magnetic effects in mixed-valence clusters were considered only for valence configuration M,(n)M(n - 1) (only metal atoms are 0166-1280:95/$09.50 #Q1995ElsevierScienceB.V. SSDZ 0166-3280(94)03830-E
All
shown for simplicity. of course a molecule also contains other atoms with the closed electronic shell). If atom recipients of the delocalised electron are paramagnetic, the delocalisation results in the double exchange phenomenon [l-3]. The double exchange effect was first introduced for magnetic semiconductors [4], and during the last decade it was applied to the mixed-valence clusters. The double exchange leads to the ferromagnetic ordering of electronic spins in the Fe(II)Fe(III) cluster [L2Fe2(p-OH)3]2+ with L = [S]. N, N’, N”-trimethyl-1,4,7-triazacyclononane The conjunction of the double exchange and vibronit interaction in three- and four-nuclear ironsulphur clusters can lead to the broken-symmetry ground state, in which the excess electron is delocalised only over a pair of centres [6,7]. Clusters in which two valence states of a transition metal correspond to electronic configurations do and d’ (e.g. W(V) and W(VI), V(IV) and V(V)) can not manifest the double exchange effects as the excesselectrons migrate between the spinlesscores.
rightsreserved
account: the resonance interaction between localised orbitals and the Coulomb repulsion of electrons at the same orbital. The physics of a system can be adequately described by the Hubbard hamiltonian [ 1l] He = C
tijalt,Uj, + UC
Fig. 1. The Keggin structure of heteropoly complexes [XOJW!~O,,]“~. Each octahedron contains a tungsten atom, the interior tetrahedron contains the heteroatom X.
But if the number of migrating electrons is more than one (and, correspondingly, the number of metal centres is more than two), the delocalisation can contribute to the stabilisation of a definite spin state. In the present communication the delocalisation mechanism of the stabilisation of spin-singlet ground states in mixed-valence clusters will be considered.
nitI$il
(1)
*
ifj.0
where u:,, ai, are the creation and annihilation operators of an electron with the spin projection cr at the ith centre, nio = a$~~,. The first term in (1) describes the electron transfer and the second term gives the energy of the intracentre repulsion. Considering one-electron orbitals diagonalising the first term in (1) one can easily see that clusters with the odd-membered rings of metal atoms (equilateral triangle, tetrahedron) feature the asymmetry of the energy spectrum relative to the sign of the transfer parameters. The orbital energies are 61 =
(2);
-t,
E? = 2t> (1)
for an equilateral El =
(3);
-t,
(2)
triangle,
t2 = 32, (1)
(3)
for a tetrahedron, t1,2 =
hht,
(1);
t2 = 0, (2)
(4)
for a square. 2. Electronic energy spectrum of mixed-valence clusters with delocalised electron pair As was pointed out above the mixed-valence clusters with the electronic configuration do . -d’ - d’ can be found for tungsten and vanadium. One can quote such compounds as the four-nuclear square-planar clusters and [W,Os(NCS)& [S]. PW&MH~W2+ the twelve-nuclear cluster [V11A~S040(HC02)]5~ [9], and the twelve-nuclear heteropolyanion with a Keggin structure (Fig. 1) in its doubly reduced “blue” state [si04wizOj6]6P [lo]. For all these compounds the ground states are spin-singlets and excited triplet states lie much higher than can be expected for the case of the ordinary antiferromagnetic exchange. If we confine ourselves to the case of one active electronic orbital at each metallic site, two basic interactions must be taken into
fl = 2(t + fl)r (1) f2.3 = gt + t1) * J9t2 +9t:
f‘j = -(t + t,), (5)
- 14tt,, (3)
(5)
for the Keggin structure (Fig. l), where two transfer parameters must be considered. The degeneracy of each level is given in parentheses after the energy value. For the cases (2) (3), and (5) the degeneracy of a ground state is defined by the sign of the electron transfer parameter, whereas for a square it is independent of the sign oft. If the ground oneelectron state g is non-degenerate then the ground state of the same system with two electrons must be spin-singlet. Due to a large enough energy gap the configuration g2 gives the main contribution in its wave function. This conclusion is confirmed by the model calculations of the square clusters of vanadium and tungsten [ 12,131. The Hund tendency to
S..4. Bor.rhch/J.
Mol.
Swuct.
the stabilisation of the spin-triplet state can win if the ground one-electron state is degenerate. Clusters with the Keggin structure have the total tetrahedral symmetry with an equilateral triangle of metal centres at each sommet of the tetrahedron. The calculations with the Hubbard hamiltonian and the VB basis result in a triplet or degenerate triplet + singlet ground state if at least one of the transfer parameters is positive [14]. If both t and ti are negative the ground state is spin-singlet. The eigenfunction of this ground state contains comparable contributions of all possible electronic pairs ij. So the average distance between electrons in a pair must be larger than the smallest metal-metal distance. This result is consistent with the experimental data. The antiferromagnetic exchange can not give very strong stabilisation of the spin-singlet at such distances. The delocalisation mechanism also allows us to explain magnetic properties of heteropoly blues in which the central ion is paramagnetic [1.5]. If the tungsten-oxygen framework contains only one non-paired electron the antiferromagnetism is observed between it and the central paramagnetic ion. In contrast this interaction is not displayed in clusters with an electronic singlet pair. This behaviour can be expected as the electron coupling caused by the delocalisation is much stronger than the antiferromagnetic exchange.
3. Vibronic stabilisation
of spin-singlet
states
As we have seen, the negative sign of the effective electron transfer parameter is needed to have a spin-singlet ground state in molecular systems with triangular rings of metal ions. The simple overlap considerations may not be valid for complicated systems such as heteropoly complexes, and the laborious quantum chemical calculations have to be done. We would like, however, to propose a new mechanism which leads to the negative sign of the effective transfer parameter even if the electronic matrix element of resonance interaction is positive. The d-electrons of transition metals in the oxygen surrounding are usually strongly coupled with local ligand-shell distortions. It is clear that in purely electronic models the actual parameters
(Throchem)
330 (1995j
141
139-143
must be considered as renormalised by the local electron-vibrational coupling. This interaction can be accounted for by the well-known Peierls~Hubbard hamiltonian H = H, + hw c
b’bi + X x(b; i
+ bi)ni,
(6)
i,rr
where b+ and hi are the creation and annihilation operators of local vibrational quanta, w being then frequency, and X describes the strength of the electron-vibrational coupling. The analysis of the energy spectrum of the hamiltonian (6) for a triangular cluster with two electrons is given elsewhere [ 161. We shall cite here some results. If the first term in (1) is taken as a perturbation, the linear approximation gives the well-known expression for the polaron-reduced electron transfer t + tl = texp (-Eh/hw).
(7)
where Eh = X2/L is the polaron binding energy. The second-order terms of the perturbation theory describe the electron transfer through intermediate virtual excited states. There are two types of such excited states: electronically excited (and their vibrational satellites) and vibrationally excited. In the electronically excited states one of the metal centres is doubly populated. From this excited state the electron can either go back to its initial site, or move to an initially empty site. The first process does not contribute to the electron transfer. It leads to the usual antiferromagnetic exchange. The real electron transfer appears only due to the second process. which was called “exchange transfer” [17]. It is clear that both effects exist only for the singlet electronic pair. The equilibrium metal-ligand distances are different for a centre with and without an electron. Thus due to non-orthogonality of the corresponding vibrational wave functions the second-order electron transfer is possible through vibrationally excited states. This term is active for both singlet and triplet electronic pairs, and due to the increasing overlap for the excited vibrational states it can even be superior to the first-order term (7). Both second-order transfer terms are negative. If for positive t the linear transfer is compensated by the “vibronic transfer”, the exchange transfer
142
S.A. BorshchjJ.
Mol.
Sfruct.
(Theochem)
330 1199.51 139-143
Fig. 2. The effective electron transfer matrix elements versus the electron-vibrational coupling (X0 = Xl&, t/hw = 2. U/~~LI = 15): 1 linear term + second-order vibronic transfer and 2 ~ second-order exchange transfer.
becomes the leading delocalisation mechanism (Fig. 2). It stabilises the singlet ground state of the electronic pair. 4. Conclusion The simple Hubbard-type model was seen to describe the behaviour of electronic pairs in the polynuclear mixed-valence compounds. Within this model the stabilisation of the spin-singlet excited state can be understood as arising from the combination of two effects: the delocalisation of electrons and the electron-vibrational coupling. Recently the resonating-valence-bond (RVB) state introduced to explain the oxide superconductors was considered to originate from singlet pair delocalisation within triangular fragments of a square lattice [18]. Mixed-valence clusters with a delocalised electronic pair could give a molecular model of such strongly correlated solids.
References [l] G. Blondin and J.-J. Girerd. Chem. Re\.. 90 (1990) 1359. [2] I.B. Bersuker and S.A. Borshch, in I. PrigopIne and S.A. Rice (Eds.), Advances in Chemical Phys I:S, Vol. 81, Wiley, New York, 1992, p. 703. [3] G. Blondin, S.A. Borshch and J.-J. Girerd, Comm. Inorg. Chem., 12 (1992) 315. [4] C. Zener, Phys. Rev., 82 (1951) 403. [5] X.Q. Ding, E.L. Bominaar, E. Bill, H. Winkl,r, A.X. Trautwein, S. Driieke, P. Chaudhuri and K. Wtrghardt, J. Chem. Phys., 92 (1990) 178. [6] S.A. Borshch. E.L. Bominaar, G. Blondin :‘rrd J.-J. Girerd, J. Am. Chem. Sot. 115 (1993) 5155. [7] E.L. Bominaar, S.A. Borshch and J.-J. Glrertl. J. .4m. Chem. Sot.. 116 (1994) 5362. [S] Y. Jeannin, J.-P. Launay, J. Livage and A. Nel, Inorg. Chem., 17 (1978) 374. [9] A. Miiller, J. Dijring and H. Biigge, J. Chcm. Sot. Chem. Commun., 1991, 273. [lo] M. Kozik. N. Casari-Pastor. C.F. Hammer anci L.C.W. Baker, J. Am. Chem. Sot., 110 (1988) 7697.
S.A. Borshch:J.
Mol.
Struct.
[I l] J. Hubbard, Proc. Roy. Sot. A, 276 (1963) 238. [12] J.-J. Girerd and J.-P. Launay, Chem. Phys., 74 (1983) 217. [13] D. Gatteschi and B.S. Tsukerblat, Molec. Phys., 79 (1993) 121. [14] S.A. Borshch and B. Bigot, Chem. Phys. Lett.. 212 (1993) 398.
(Theo&em)
330 (199.5)
139-143
143
[15] N. Casaii-Pastor and L.C.W. Baker, J. Am. Chem Sot., 114 (1992) 10384. [16] S.A. Borshch and J.-J. Girerd, Chem. Phys., 181 (1994) I. [17] G. Blondin and J.-J. Girerd, in K. Prassides (Ed.). Mixed Valency Systems: Applications in Chemistry, Physics and Biology, Kluwer, Dordrecht. 1991, p. 353. [18] K. Takano and K. Sano. Phys. Rev. B. 39 (1989) 7367.
Journal
of Molecular
Structure
(Theochem)
330 (1995)
139-143
Stabilisation of spin-singlet electronic states in transition metal clusters with a delocalised electron pair S.A. Borshch Received
24 September
1993; accepted
27 September
1993
Abstract
A modelis proposedfor the interpretation of magneticpropertiesof polynuclearmixed-valencecomplexescontaining a pair of delocalisedelectrons.It is shownthat in many casesthe delocalisationleadsto a spin-singletground stateof clusters.The combineddelocalisation+ vibronic mechanismof stabilisationof the spin-singletstate is proposedfor clusterscontaining trianglesof metal atoms.
1. Introduction The magnetic properties of molecular systems containing paramagnetic ions in most cases are determined by localised unpaired electrons. Polynuclear clusters of transition metals present analogies, on a molecular level, to solid-state magnetic dielectrics. However, over the past few years the synthetic chemistry has provided more and more molecular systems in which the unpaired electrons can migrate between metallic centres. First of all we mean the mixed-valence compounds containing atoms of the same metal in different valence states. If the surroundings of metal atoms are equivalent chemically and structurally, the conditions are especially favourable for the delocalisation of the excess electrons. Thus mixed-valence clusters can be considered as molecular analogues of solids with itinerant electrons. Until now the magnetic effects in mixed-valence clusters were considered only for valence configuration M,(n)M(n - 1) (only metal atoms are 0166-1280:95/$09.50 #Q1995ElsevierScienceB.V. SSDZ 0166-3280(94)03830-E
All
shown for simplicity. of course a molecule also contains other atoms with the closed electronic shell). If atom recipients of the delocalised electron are paramagnetic, the delocalisation results in the double exchange phenomenon [l-3]. The double exchange effect was first introduced for magnetic semiconductors [4], and during the last decade it was applied to the mixed-valence clusters. The double exchange leads to the ferromagnetic ordering of electronic spins in the Fe(II)Fe(III) cluster [L2Fe2(p-OH)3]2+ with L = [S]. N, N’, N”-trimethyl-1,4,7-triazacyclononane The conjunction of the double exchange and vibronit interaction in three- and four-nuclear ironsulphur clusters can lead to the broken-symmetry ground state, in which the excess electron is delocalised only over a pair of centres [6,7]. Clusters in which two valence states of a transition metal correspond to electronic configurations do and d’ (e.g. W(V) and W(VI), V(IV) and V(V)) can not manifest the double exchange effects as the excesselectrons migrate between the spinlesscores.
rightsreserved
account: the resonance interaction between localised orbitals and the Coulomb repulsion of electrons at the same orbital. The physics of a system can be adequately described by the Hubbard hamiltonian [ 1l] He = C
tijalt,Uj, + UC
Fig. 1. The Keggin structure of heteropoly complexes [XOJW!~O,,]“~. Each octahedron contains a tungsten atom, the interior tetrahedron contains the heteroatom X.
But if the number of migrating electrons is more than one (and, correspondingly, the number of metal centres is more than two), the delocalisation can contribute to the stabilisation of a definite spin state. In the present communication the delocalisation mechanism of the stabilisation of spin-singlet ground states in mixed-valence clusters will be considered.
nitI$il
(1)
*
ifj.0
where u:,, ai, are the creation and annihilation operators of an electron with the spin projection cr at the ith centre, nio = a$~~,. The first term in (1) describes the electron transfer and the second term gives the energy of the intracentre repulsion. Considering one-electron orbitals diagonalising the first term in (1) one can easily see that clusters with the odd-membered rings of metal atoms (equilateral triangle, tetrahedron) feature the asymmetry of the energy spectrum relative to the sign of the transfer parameters. The orbital energies are 61 =
(2);
-t,
E? = 2t> (1)
for an equilateral El =
(3);
-t,
(2)
triangle,
t2 = 32, (1)
(3)
for a tetrahedron, t1,2 =
hht,
(1);
t2 = 0, (2)
(4)
for a square. 2. Electronic energy spectrum of mixed-valence clusters with delocalised electron pair As was pointed out above the mixed-valence clusters with the electronic configuration do . -d’ - d’ can be found for tungsten and vanadium. One can quote such compounds as the four-nuclear square-planar clusters and [W,Os(NCS)& [S]. PW&MH~W2+ the twelve-nuclear cluster [V11A~S040(HC02)]5~ [9], and the twelve-nuclear heteropolyanion with a Keggin structure (Fig. 1) in its doubly reduced “blue” state [si04wizOj6]6P [lo]. For all these compounds the ground states are spin-singlets and excited triplet states lie much higher than can be expected for the case of the ordinary antiferromagnetic exchange. If we confine ourselves to the case of one active electronic orbital at each metallic site, two basic interactions must be taken into
fl = 2(t + fl)r (1) f2.3 = gt + t1) * J9t2 +9t:
f‘j = -(t + t,), (5)
- 14tt,, (3)
(5)
for the Keggin structure (Fig. l), where two transfer parameters must be considered. The degeneracy of each level is given in parentheses after the energy value. For the cases (2) (3), and (5) the degeneracy of a ground state is defined by the sign of the electron transfer parameter, whereas for a square it is independent of the sign oft. If the ground oneelectron state g is non-degenerate then the ground state of the same system with two electrons must be spin-singlet. Due to a large enough energy gap the configuration g2 gives the main contribution in its wave function. This conclusion is confirmed by the model calculations of the square clusters of vanadium and tungsten [ 12,131. The Hund tendency to
S..4. Bor.rhch/J.
Mol.
Swuct.
the stabilisation of the spin-triplet state can win if the ground one-electron state is degenerate. Clusters with the Keggin structure have the total tetrahedral symmetry with an equilateral triangle of metal centres at each sommet of the tetrahedron. The calculations with the Hubbard hamiltonian and the VB basis result in a triplet or degenerate triplet + singlet ground state if at least one of the transfer parameters is positive [14]. If both t and ti are negative the ground state is spin-singlet. The eigenfunction of this ground state contains comparable contributions of all possible electronic pairs ij. So the average distance between electrons in a pair must be larger than the smallest metal-metal distance. This result is consistent with the experimental data. The antiferromagnetic exchange can not give very strong stabilisation of the spin-singlet at such distances. The delocalisation mechanism also allows us to explain magnetic properties of heteropoly blues in which the central ion is paramagnetic [1.5]. If the tungsten-oxygen framework contains only one non-paired electron the antiferromagnetism is observed between it and the central paramagnetic ion. In contrast this interaction is not displayed in clusters with an electronic singlet pair. This behaviour can be expected as the electron coupling caused by the delocalisation is much stronger than the antiferromagnetic exchange.
3. Vibronic stabilisation
of spin-singlet
states
As we have seen, the negative sign of the effective electron transfer parameter is needed to have a spin-singlet ground state in molecular systems with triangular rings of metal ions. The simple overlap considerations may not be valid for complicated systems such as heteropoly complexes, and the laborious quantum chemical calculations have to be done. We would like, however, to propose a new mechanism which leads to the negative sign of the effective transfer parameter even if the electronic matrix element of resonance interaction is positive. The d-electrons of transition metals in the oxygen surrounding are usually strongly coupled with local ligand-shell distortions. It is clear that in purely electronic models the actual parameters
(Throchem)
330 (1995j
141
139-143
must be considered as renormalised by the local electron-vibrational coupling. This interaction can be accounted for by the well-known Peierls~Hubbard hamiltonian H = H, + hw c
b’bi + X x(b; i
+ bi)ni,
(6)
i,rr
where b+ and hi are the creation and annihilation operators of local vibrational quanta, w being then frequency, and X describes the strength of the electron-vibrational coupling. The analysis of the energy spectrum of the hamiltonian (6) for a triangular cluster with two electrons is given elsewhere [ 161. We shall cite here some results. If the first term in (1) is taken as a perturbation, the linear approximation gives the well-known expression for the polaron-reduced electron transfer t + tl = texp (-Eh/hw).
(7)
where Eh = X2/L is the polaron binding energy. The second-order terms of the perturbation theory describe the electron transfer through intermediate virtual excited states. There are two types of such excited states: electronically excited (and their vibrational satellites) and vibrationally excited. In the electronically excited states one of the metal centres is doubly populated. From this excited state the electron can either go back to its initial site, or move to an initially empty site. The first process does not contribute to the electron transfer. It leads to the usual antiferromagnetic exchange. The real electron transfer appears only due to the second process. which was called “exchange transfer” [17]. It is clear that both effects exist only for the singlet electronic pair. The equilibrium metal-ligand distances are different for a centre with and without an electron. Thus due to non-orthogonality of the corresponding vibrational wave functions the second-order electron transfer is possible through vibrationally excited states. This term is active for both singlet and triplet electronic pairs, and due to the increasing overlap for the excited vibrational states it can even be superior to the first-order term (7). Both second-order transfer terms are negative. If for positive t the linear transfer is compensated by the “vibronic transfer”, the exchange transfer
142
S.A. BorshchjJ.
Mol.
Sfruct.
(Theochem)
330 1199.51 139-143
Fig. 2. The effective electron transfer matrix elements versus the electron-vibrational coupling (X0 = Xl&, t/hw = 2. U/~~LI = 15): 1 linear term + second-order vibronic transfer and 2 ~ second-order exchange transfer.
becomes the leading delocalisation mechanism (Fig. 2). It stabilises the singlet ground state of the electronic pair. 4. Conclusion The simple Hubbard-type model was seen to describe the behaviour of electronic pairs in the polynuclear mixed-valence compounds. Within this model the stabilisation of the spin-singlet excited state can be understood as arising from the combination of two effects: the delocalisation of electrons and the electron-vibrational coupling. Recently the resonating-valence-bond (RVB) state introduced to explain the oxide superconductors was considered to originate from singlet pair delocalisation within triangular fragments of a square lattice [18]. Mixed-valence clusters with a delocalised electronic pair could give a molecular model of such strongly correlated solids.
References [l] G. Blondin and J.-J. Girerd. Chem. Re\.. 90 (1990) 1359. [2] I.B. Bersuker and S.A. Borshch, in I. PrigopIne and S.A. Rice (Eds.), Advances in Chemical Phys I:S, Vol. 81, Wiley, New York, 1992, p. 703. [3] G. Blondin, S.A. Borshch and J.-J. Girerd, Comm. Inorg. Chem., 12 (1992) 315. [4] C. Zener, Phys. Rev., 82 (1951) 403. [5] X.Q. Ding, E.L. Bominaar, E. Bill, H. Winkl,r, A.X. Trautwein, S. Driieke, P. Chaudhuri and K. Wtrghardt, J. Chem. Phys., 92 (1990) 178. [6] S.A. Borshch. E.L. Bominaar, G. Blondin :‘rrd J.-J. Girerd, J. Am. Chem. Sot. 115 (1993) 5155. [7] E.L. Bominaar, S.A. Borshch and J.-J. Glrertl. J. .4m. Chem. Sot.. 116 (1994) 5362. [S] Y. Jeannin, J.-P. Launay, J. Livage and A. Nel, Inorg. Chem., 17 (1978) 374. [9] A. Miiller, J. Dijring and H. Biigge, J. Chcm. Sot. Chem. Commun., 1991, 273. [lo] M. Kozik. N. Casari-Pastor. C.F. Hammer anci L.C.W. Baker, J. Am. Chem. Sot., 110 (1988) 7697.
S.A. Borshch:J.
Mol.
Struct.
[I l] J. Hubbard, Proc. Roy. Sot. A, 276 (1963) 238. [12] J.-J. Girerd and J.-P. Launay, Chem. Phys., 74 (1983) 217. [13] D. Gatteschi and B.S. Tsukerblat, Molec. Phys., 79 (1993) 121. [14] S.A. Borshch and B. Bigot, Chem. Phys. Lett.. 212 (1993) 398.
(Theo&em)
330 (199.5)
139-143
143
[15] N. Casaii-Pastor and L.C.W. Baker, J. Am. Chem Sot., 114 (1992) 10384. [16] S.A. Borshch and J.-J. Girerd, Chem. Phys., 181 (1994) I. [17] G. Blondin and J.-J. Girerd, in K. Prassides (Ed.). Mixed Valency Systems: Applications in Chemistry, Physics and Biology, Kluwer, Dordrecht. 1991, p. 353. [18] K. Takano and K. Sano. Phys. Rev. B. 39 (1989) 7367.